Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include <bits/stdc++.h>

const int MAX_N = 3000;

int n;
double f[MAX_N][MAX_N];

int main()
{
	std::cin >> n;
	f[0][0] = 1.0;
	for (int i = 0; i < n; ++i)
	{
		double p; std::cin >> p;
		for (int j = 0; j <= i; ++j)
		{
			f[i + 1][j + 1] += f[i][j] * p;
			f[i + 1][  j  ] += f[i][j] * (1.0 - p);
		}
	}
	std::cout << std::fixed << std::setprecision(10);
	std::cout << std::accumulate(f[n] + n / 2 + 1, f[n] + n + 1, 0.0) << '\n';
}